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Research Article Open access CC BY 4.0

The \(\alpha\)-analogues of r-Whitney Numbers via Normal Ordering

Hye Kyung Kim

Journal of Advances in Mathematics and Computer Science · pp. 33–50 · Published 17 Nov 2022

10.9734/jamcs/2022/v37i101716

Abstract

The normal ordering of an integral power of the number operator a\(^\dagger\)a in terms of boson annihilation a and creation a\(^\dagger\) operators is expressed with the help of the Stirling numbers of the second kind. The normal ordering problems directly links the problems to combinatorics. Whit this in mind, in this paper, we define the \(\alpha\)-analogues of r-Whitney numbers of the first kind and those of second kind, which are different from degenerate r-Whitney numbers. We show that \(\alpha\)-analogues falling factorial of the number operator is expressed in terms of the \(\alpha\)-analogues of r-Whitney numbers of the first kind and its inverse formula is expressed as those of the second kind. We also derived some properties, recurrence relations and several identities on those numbers arising from Boson annihilation a and creation operators a\(^\dagger\), number operators \(\hat{h}\) and coherent states.

\(\alpha\)-analogues of r-Stirling numbers of the first kind \(\alpha\)-analogues of r-Stirling numbers of the second kind r-Whitney numbers of the first kind r-Whitney numbers of the second kind Normal ordering

References (20)

  1. 1 Advanced Combinatorics: The Art of Finite and Infinite Expansions [DOI]
  2. 2 A New Formula for the Bernoulli Polynomials [DOI]
  3. 3 On Whitney numbers of Dowling lattices [DOI]
  4. 4 r -Whitney numbers of Dowling lattices [DOI]
  5. 5 On Some Numbers Related to Whitney Numbers of Dowling Lattices [DOI]
  6. 6 Some Identities on Degenerate $$r$$-Stirling Numbers via Boson Operators [DOI]
  7. 7 Translated Whitney and r-Whitney Numbers: A Combinatorial Approach
  8. 8 Degenerate r-Whitney numbers and degenerate r-Dowling polynomials via boson operators [DOI]
  9. 9 A note on degenerate r-Stirling numbers [DOI]
  10. 10 Combinatorial aspects of boson algebra [DOI]
  11. 11 Bell numbers and coherent states [DOI]
  12. 12 Normal ordering of degenerate integral powers of number operator and its applications [DOI]
  13. 13 A comprehensive study of $${\varvec{r}}$$ r -Dowling polynomials [DOI]
  14. 14 New combinatorial interpretations of r -Whitney and r -Whitney–Lah numbers [DOI]
  15. 15 The Translated Dowling Polynomials and Numbers [DOI]
  16. 16 Some identities involving degenerate Stirling numbers arising from normal ordering [DOI]
  17. 17 Degenerater‐Bell Polynomials Arising from Degenerate Normal Ordering [DOI]
  18. 18 Coherent states and combinatorics [DOI]
  19. 19 A note on degenerate stirling polynomials of the second kind [DOI]
  20. 20 Some identities on $λ$-analogues of $r$-stirling numbers of the second kind [DOI]

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